_0^ 2 / 3 d x 4+9 x^2 = - Vidyadip

MCQ

$\int_0^{2 / 3} \frac{d x}{4+9 x^2}=$

A
$\frac{\pi}{12}$
✓
$\frac{\pi}{24}$
C
$\frac{\pi}{4}$
D
$0$

✓

Answer

Correct option: B.

$\frac{\pi}{24}$

(B)
$\int_0^{2 / 3} \frac{d x}{4+9 x^2}=\frac{1}{9} \int_0^{\frac{2}{3}} \frac{d x}{\left(\frac{2}{3}\right)^2+x^2}$
$=\frac{1}{9} \times \frac{1}{\frac{2}{3}}\left[\tan ^{-1} \frac{x}{\frac{2}{3}}\right]_0^{\frac{2}{3}}$
$=\frac{1}{6} \times \frac{\pi}{4}=\frac{\pi}{24}$

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